non-Euclidean Distance Measures Research Articles

Unsupervised learning methods for efficient geographic clustering and identification of disease disparities with applications to county-level colorectal cancer incidence in California

Many public health policymaking questions involve data subsets representing application-specific attributes and geographic location. We develop and evaluate standard and tailored techniques for clustering via unsupervised learning (UL) algorithms on such amalgamated (dual-domain) data sets. The aim of the associated algorithms is to identify geographically efficient clusters that also maximize the number of statistically significant differences in disease incidence and demographic variables across top clusters. Two standard UL approaches, k means with k++ initialization (k++) and the standard self-organizing map (SSOM), are considered along with a new, tailored version of the SOM (TSOM). The TSOM algorithm involves optimization of a customized objective function with terms promoting individual geographic cluster cohesion while also maximizing the number of differences across clusters, and two hyper-parameters controlling the relative weighting of geographic and attribute subspaces in a non-Euclidean distance measure within the clustering problem. The performance of these three techniques (k++, SSOM, TSOM) is compared and evaluated in the context of a data set for colorectal cancer incidence in the state of California, at the level of individual counties. Clusters are visualized via chloropleth maps and ordered graphs are also used to illustrate disparities in disease incidence among four identity groups. While all three approaches performed well, the TSOM identified the largest number of disease and demographic disparities while also yielding more geographically efficient top clusters. Techniques presented in this study are relevant to applications including the delivery of health care resources and identifying disparities among identity groups, and to questions involving coordination between county- and state-level policymakers.

Non-Euclidean distance measures in spatial data decision analysis: investigations for mineral potential mapping

As a powerful tool for decision analysis, fifteen distance measures are incorporated into the technique for order preference by similarity to ideal solution (TOPSIS) to carry out a knowledge-driven approach in mineral exploration, whereby a multi-criterion decision making problem is solved for spatial data analysis. The kernel of the original version of the TOPSIS method as an integral part of the analysis will be defined on Euclidean distance. This research investigates to define the kernel on each of 15 distance measures from four general families of functions including L1, intersection, inner product and fidelity. To check the performance of each distance measure, the North Narbaghi porphyry copper mine in Iran was chosen. The significance of kernel substitution lies in the improvement of synthesized evidence maps in comparison to the Euclidean-based TOPSIS method in this study.

Kernel-Based Robust Bias-Correction Fuzzy Weighted C-Ordered-Means Clustering Algorithm

The spatial constrained Fuzzy C-means clustering (FCM) is an effective algorithm for image segmentation. Its background information improves the insensitivity to noise to some extent. In addition, the membership degree of Euclidean distance is not suitable for revealing the non-Euclidean structure of input data, since it still lacks enough robustness to noise and outliers. In order to overcome the problem above, this paper proposes a new kernel-based algorithm based on the Kernel-induced Distance Measure, which we call it Kernel-based Robust Bias-correction Fuzzy Weighted C-ordered-means Clustering Algorithm (KBFWCM). In the construction of the objective function, KBFWCM algorithm comprehensively takes into account that the spatial constrained FCM clustering algorithm is insensitive to image noise and involves a highly intensive computation. Aiming at the insensitivity of spatial constrained FCM clustering algorithm to noise and its image detail processing, the KBFWCM algorithm proposes a comprehensive algorithm combining fuzzy local similarity measures (space and grayscale) and the typicality of data attributes. Aiming at the poor robustness of the original algorithm to noise and outliers and its highly intensive computation, a Kernel-based clustering method that includes a class of robust non-Euclidean distance measures is proposed in this paper. The experimental results show that the KBFWCM algorithm has a stronger denoising and robust effect on noise image.

Stochastic matched-field localization of an acoustic source based on principles of Riemannian geometry.

Passive localization of acoustic sources is treated within a geometric framework where non-Euclidean distance measures are computed between a cross-spectral density estimate of received data on a vertical array and a set of stochastic replica steering matrices, rather than traditional replica steering vectors. A processing scheme involving matrix-matrix comparisons where steering matrices, as functions of the replica source coordinates, naturally incorporate environmental variability or uncertainty provides a general framework for considering the acoustic inverse source problem in an ocean waveguide. Within this context a subset of matched-field processors is examined, based on recent advances in the application of non-Euclidean geometry to statistical classification of data feature clusters. The matrices are interpreted abstractly as points in a Riemannian manifold, and an appropriately defined distance measure between pairs of matrices on this manifold defines a matched-field processor for estimating source location. Acoustic simulations are performed for a waveguide comprising both a depth-dependent sound-speed profile perturbed by linear internal gravity waves and a depth-correlated surface noise field, providing an example of the viability of this approach to passive source localization in the presence of sound-speed variability.

Faster Algorithms for the Constrained k-means Problem

The classical center based clustering problems such as k-means/median/center assume that the optimal clusters satisfy the locality property that the points in the same cluster are close to each other. A number of clustering problems arise in machine learning where the optimal clusters do not follow such a locality property. For instance, consider the r -gather clustering problem where there is an additional constraint that each of the clusters should have at least r points or the capacitated clustering problem where there is an upper bound on the cluster sizes. Consider a variant of the k-means problem that may be regarded as a general version of such problems. Here, the optimal clusters O 1, ..., O k are an arbitrary partition of the dataset and the goal is to output k-centers c 1, ..., c k such that the objective function ${\sum }_{i = 1}^{k} {\sum }_{x \in O_{i}} ||x - c_{i}||^{2}$ is minimized. It is not difficult to argue that any algorithm (without knowing the optimal clusters) that outputs a single set of k centers, will not behave well as far as optimizing the above objective function is concerned. However, this does not rule out the existence of algorithms that output a list of such k centers such that at least one of these k centers behaves well. Given an error parameter e > 0, let l denote the size of the smallest list of k-centers such that at least one of the k-centers gives a (1 + e) approximation w.r.t. the objective function above. In this paper, we show an upper bound on l by giving a randomized algorithm that outputs a list of $2^{\tilde {O}(k/\varepsilon )}$ k-centers. We also give a closely matching lower bound of $2^{\tilde {\Omega }(k/\sqrt {\varepsilon })}$ . Moreover, our algorithm runs in time $O \left (n d \cdot 2^{\tilde {O}(k/\varepsilon )} \right )$ . This is a significant improvement over the previous result of Ding and Xu (2015) who gave an algorithm with running time O(n d ⋅ (log n) k ⋅ 2 p o l y(k/e)) and output a list of size O((log n) k ⋅ 2 p o l y(k/e)). Our techniques generalize for the k-median problem and for many other settings where non-Euclidean distance measures are involved.

Effective FCM noise clustering algorithms in medical images

The main motivation of this paper is to introduce a class of robust non-Euclidean distance measures for the original data space to derive new objective function and thus clustering the non-Euclidean structures in data to enhance the robustness of the original clustering algorithms to reduce noise and outliers. The new objective functions of proposed algorithms are realized by incorporating the noise clustering concept into the entropy based fuzzy C-means algorithm with suitable noise distance which is employed to take the information about noisy data in the clustering process. This paper presents initial cluster prototypes using prototype initialization method, so that this work tries to obtain the final result with less number of iterations. To evaluate the performance of the proposed methods in reducing the noise level, experimental work has been carried out with a synthetic image which is corrupted by Gaussian noise. The superiority of the proposed methods has been examined through the experimental study on medical images. The experimental results show that the proposed algorithms perform significantly better than the standard existing algorithms. The accurate classification percentage of the proposed fuzzy C-means segmentation method is obtained using silhouette validity index.

Analytical methods for quantifying environmental connectivity for the control and surveillance of infectious disease spread

The sustained transmission and spread of environmentally mediated infectious diseases is governed in part by the dispersal of parasites, disease vectors and intermediate hosts between sites of transmission. Functional geospatial models can be used to quantify and predict the degree to which environmental features facilitate or limit connectivity between target populations, yet typical models are limited in their geographical and analytical approach, providing simplistic, global measures of connectivity and lacking methods to assess the epidemiological implications of fine-scale heterogeneous landscapes. Here, functional spatial models are applied to problems of surveillance and control of the parasitic blood fluke Schistosoma japonicum and its intermediate snail host Oncomelania haupensis in western China. We advance functional connectivity methods by providing an analytical framework to (i) identify nodes of transmission where the degree of connectedness to other villages, and thus the potential for disease spread, is higher than is estimated using Euclidean distance alone and (ii) (re)organize transmission sites into disease surveillance units based on second-order relationships among nodes using non-Euclidean distance measures, termed effective geographical distance (EGD). Functional environmental models are parametrized using ecological information on the target organisms, and pair-wise distributions of inter-node EGD are estimated. A Monte Carlo rank product analysis is presented to identify nearby nodes under alternative distance models. Nodes are then iteratively embedded into EGD space and clustered using a k-means algorithm to group villages into ecologically meaningful surveillance groups. A consensus clustering approach is taken to derive the most stable cluster structure. The results indicate that novel relationships between nodes are revealed when non-Euclidean, ecologically determined distance measures are used to quantify connectivity in heterogeneous landscapes. These connections are not evident when analysing nodes in Euclidean space, and thus surveillance and control activities planned using Euclidean distance measures may be suboptimal. The methods developed here provide a quantitative framework for assessing the effectiveness of ecologically grounded surveillance systems and of control and prevention strategies for environmentally mediated diseases.

Clustering: A neural network approach

Clustering is a fundamental data analysis method. It is widely used for pattern recognition, feature extraction, vector quantization (VQ), image segmentation, function approximation, and data mining. As an unsupervised classification technique, clustering identifies some inherent structures present in a set of objects based on a similarity measure. Clustering methods can be based on statistical model identification (McLachlan & Basford, 1988) or competitive learning. In this paper, we give a comprehensive overview of competitive learning based clustering methods. Importance is attached to a number of competitive learning based clustering neural networks such as the self-organizing map (SOM), the learning vector quantization (LVQ), the neural gas, and the ART model, and clustering algorithms such as the C-means, mountain/subtractive clustering, and fuzzy C-means (FCM) algorithms. Associated topics such as the under-utilization problem, fuzzy clustering, robust clustering, clustering based on non-Euclidean distance measures, supervised clustering, hierarchical clustering as well as cluster validity are also described. Two examples are given to demonstrate the use of the clustering methods.

On the Use of Non-Euclidean Distance Measures in Geostatistics

In many scientific disciplines, straight line, Euclidean distances may not accurately describe proximity relationships among spatial data. However, non-Euclidean distance measures must be used with caution in geostatistical applications. A simple example is provided to demonstrate there are no guarantees that existing covariance and variogram functions remain valid (i.e. positive definite or conditionally negative definite) when used with a non-Euclidean distance measure. There are certain distance measures that when used with existing covariance and variogram functions remain valid, an issue that is explored. The concept of isometric embedding is introduced and linked to the concepts of positive and conditionally negative definiteness to demonstrate classes of valid norm dependent isotropic covariance and variogram functions, results many of which have yet to appear in the mainstream geostatistical literature or application. These classes of functions extend the well known classes by adding a parameter to define the distance norm. In practice, this distance parameter can be set a priori to represent, for example, the Euclidean distance, or kept as a parameter to allow the data to choose the metric. A simulated application of the latter is provided for demonstration. Simulation results are also presented comparing kriged predictions based on Euclidean distance to those based on using a water metric.

Integrating the improved CBP model with kernel SOM

In this paper, we first design a more generalized network model, Improved circular back propagation (CBP), based on the same structure as CBP proposed by Ridella et al. The novelty of Improved circular back propagation neural network (ICBP) lies in: (1) it substitutes the original extra added node with the isotropic quadratic form input in CBP with the one with an anisotropic quadratic form input; (2) particularly, the weights between the extra node and all the hidden nodes are endowed fixed values instead of the original changeable values. As a result, ICBP possesses better generalization and adaptability although it has less adjustable weights compared to CBP. Secondly, we propose a new kernel-based self-organizing maps (KSOM) algorithm using the kernel method. Our main motives of using the kernel method are (1) to induce a class of robust non-Euclidean distance measures for the original input space and establish a new objective functions for self-organizing maps (SOM), and thus make the newly established SOM able to cluster the non-Euclidean structures in data; (2) to enhance robustness of the SOM algorithms to noise and outliers and at the same time still retain computational simplicity. And then, with the combined BP-SOM idea of Weijters, we construct a new integrated network ICBP–KSOM. Our motivation of presenting the integration is to construct a high-performance classifier by utilizing both ICBP's good generalization and adaptability and KSOM's higher classification performance and robustness comparing to SOM. Finally, the experimental results on three benchmark data sets show the superiority and effectiveness of our new integration in terms of the t-test.

Mining Navigation Patterns Using a Sequence Alignment Method

In this article, a new method is illustrated for mining navigation patterns on a web site. Instead of clustering patterns by means of a Euclidean distance measure, in this approach users are partitioned into clusters using a non-Euclidean distance measure called the Sequence Alignment Method (SAM). This method partitions navigation patterns according to the order in which web pages are requested and handles the problem of clustering sequences of different lengths. The performance of the algorithm is compared with the results of a method based on Euclidean distance measures. SAM is validated by means of user-traffic data of two different web sites. Empirical results show that SAM identifies sequences with similar behavioral patterns not only with regard to content, but also considering the order of pages visited in a sequence.

A computational feature binding model of human texture perception

We present a computational model for human texture perception which assigns functional principles to the Gestalt laws of similarity and proximity. Motivated by early vision mechanisms, in the first stage, local texture features are extracted by utilizing multi-scale filtering and nonlinear spatial pooling. In the second stage, features are grouped according to the spatial feature binding model of the competitive layer model (CLM; Wersing et al. 2001). The CLM uses cooperative and competitive interactions in a recurrent network, where binding is expressed by the layer-wise coactivation of feature-representing neurons. The Gestalt law of similarity is expressed by a non-Euclidean distance measure in the abstract feature space with proximity being taken into account by a spatial component. To choose the stimulus dimensions which allow the most salient similarity-based texture segmentation, the feature similarity metrics is reduced to the directions of maximum variance. We show that our combined texture feature extraction and binding model performs segmentation in strong conformity with human perception. The examples range from classical microtextures and Brodatz textures to other classical Gestalt stimuli, which offer a new perspective on the role of texture for more abstract similarity grouping.

Segmentation of visiting patterns on web sites using a sequence alignment method

In this article, a new method is illustrated for segmentation of visiting patterns on a web site. Instead of clustering users by means of a Euclidean distance measure, in our approach users are partitioned into clusters using a non-Euclidean distance measure, called Sequence Alignment Method (SAM). This method ensures that sequential relationships, represented by the order of elements, are taken into account. In experiments using real traffic data on the web site of a Belgian telecom provider, the performance of SAM is compared with the results of a method based on Euclidean distance measures. Empirical results show that SAM identifies segments presenting behavioral characteristics not only with regard to content but also considering the order of pages that are visited on a web site.

Modulation and Coding for Noncoherent Communications

Until recently, the theory of noncoherent communications was premised on the use of orthogonal multi-pulse modulation such as frequency shift keying. The main drawback of this modulation scheme has been its poor spectral efficiency (rate/bandwidth). This paper considers instead the more general non-orthogonal multi-pulse modulation (NMM) technique. Optimal and suboptimal noncoherent detection strategies for NMM are reviewed and their asymptotic (high SNR) performances are characterized for the additive Gaussian as well as the Rayleigh fading channels. The resulting non-Euclidean distance measures are then used to design NMM signal sets that yield significantly higher bandwidth efficiencies than their orthogonal counterparts. NMM in conjunction with convolutional coding is also studied as a way to improve energy efficiency. Several optimal convolutional codes are examined together with our signal designs. An introduction to equalization on the noncoherent channel is also presented and illustrated by example. This paper thus contains several new results and attempts at the same time to give a tutorial exposition of the subject of noncoherent communications.

Change-vector analysis in multitemporal space: A tool to detect and categorize land-cover change processes using high temporal-resolution satellite data

Analysis of change vectors in the multitemporal space, applied to multitemporal local area coverage imagery obtained by the Advanced Very-High Resolution Radiometer on NOAA-9 and NOAA-11 orbiting platforms, clearly reveals the nature and magnitude of land-cover change in a region of West Africa. The change vector compares the difference in the time-trajectory of a biophysical indicator, such as the normalized difference vegetation index, for two successive time periods, such as hydrological years. In establishing the time-trajectory, the indicator is composited for each pixel in a registered multidate image sequence. The change vector is simply the vector difference between successive time-trajectories, each represented as a vector in a multidimensional measurement space. The length of the change vector indicates the magnitude of the interannual change, while its direction indicates the nature of the change. A principal components analysis of change vectors for a Sudanian-Sahelian region in West Africa shows four major classes of change magnitude and four general contrasting types of change. Scene-specific changes, such as reservoir water level storage changes, are also identified. The technique is easily extended to other biophysical parameters, such as surface temperature, and can incorporate noneuclidean distance measures. Change vector analysis is being developed for application to the land-cover change product to be produced using NASA's Moderate-Resolution Imaging Spectroradiometer instrument, scheduled for flight in 1998 and 2000 on EOS-AM and -PM platforms.